Quantification of polymer viscoelastic effects on sor reduction using modified capillary

ABSTRACT

A method of quantifying a viscoelastic effect of a polymer on residual oil saturation (S or) including calculating an extensional capillary number (Nce) using flux, pore-scale apparent viscosity, and interfacial tension to account for the polymer&#39;s viscoelastic forces that are responsible for Sor reduction. The polymer is used polymer flooding during enhanced oil recovery. An extensional capillary number is calculated for a plurality of polymer materials, which are then compiled in a database. Also provided is a reservoir simulator for predicting the Sor reduction potential of the viscoelastic polymer, which includes a database of calculated extensional capillary numbers for a plurality of polymers. The database includes a curve generated from the calculated extensional capillary numbers for a plurality of polymers properties, flux rates, formation nature, oil viscosities, and rheological behaviors.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority of U.S. Provisional Patent Application Ser. No. 62/879,192 filed Jul. 26, 2019, which is incorporated herein by reference.

FIELD OF INVENTION

The present invention generally relates to methods of predicting residual oil saturation (Sor), and more particularly to methods of predicting residual oil saturation during viscoelastic polymer flooding during enhanced oil recovery (EOR).

BACKGROUND

Incremental oil recovered during enhanced oil recovery (EOR) processes could be due to both enhanced sweep and displacement efficiency. Sweep efficiency will be higher if the displacing slugs contact more oil with a minimal pore volume of injection. High oil viscosity and heterogeneity are the limiting factors affecting sweep efficiency. Sweep conventionally believed to be the field-scale mechanism has been reported to be a core-scale phenomenon, especially if the oil viscosity is high. The main purpose of any EOR methods is to increase the oil recovery factor in the water flooded reservoir. The oil left in the reservoir after water flooding could be either residual oil or bypassed oil. Conventionally it is believed that polymer flood can only recover the bypassed oil by increasing the sweep efficiency. In recent times, viscoelastic polymer flooding was reported to cause an increase in Sor reduction. Capillary number conventionally used to correlate Sor reduction were reported to remain the same for polymers of various elasticity, despite their differences in the Sor reduction potential. The conventional notion is that the higher the Nc, the higher the Sor reduction.

Microscopic displacement efficiency is not a core-scale phenomenon; Rather strictly, it is a pore-scale phenomenon. Residual oil by definition is the oil that is swept well by the displacing water but that failed to become mobilized due to high interfacial tension (IFT) between the water and oil. Capillarity will be higher when the IFT is high. Smaller pore radius also leads to higher capillary forces (Green and Willhite 1998). The interplay between the viscous and capillary force has been well described by the dimensionless number called capillary number (Nc). Generally, oil will be trapped at the capillary pressure of 1000 psi/ft while the viscous force is of the order of few psi/ft (Peter 2002). This residual oil can be recovered if the viscous force provided by the displacing slugs is sufficient enough to overcome the trapping capillary force (Peter 2002). The relation between S_(or) reduction and N_(c) is described by the capillary desaturation curve (CDC) (Green and Willhite 1998). As per the CDC curve, rapid oil mobilization begins to occur when N_(c) exceeds 10⁻⁵ to 10⁻⁴ for sandstone reservoirs (Melrose and Brander 1974; Stegemeier 1974; Abrams 1975; Chatzis and Morrow 1984; Chatzis et al. 1988; Johannesen et al. 2007; Humphry et al. 2014) and complete mobilization is expected to occur only when the N_(c) exceeds 10⁻² (Foster 1973; Abrams 1975; Chatzis and Morrow 1984; Jr. et al. 1985). To increase the S_(or) reduction, N_(c) has to be increased. Mathematically, viscous force can be increased by either increasing the injection rate or by increasing the displacing fluid's viscosity. Both the options are not practically feasible, as increasing the viscosity by several orders may lead to injectivity issues and increasing the rate leads to the fracturing of reservoirs. Most of the residual oil is located in the farthest part of the wellbore, where low flux conditions prevail. The only way to recover residual oil that is practically feasible is to reduce the capillary trapping force for which the surfactant flooding capable of providing ultra-low IFT is employed (Green and Willhite 1998). IFT has been used a parameter to screen or formulate the optimal surfactants for enhancing microscopic oil recovery (Azad 2014; Azad and Sultan 2014; Han et al. 2013; Kamal et al. 2017).

Polymer solutions are conventionally employed to increase the sweep efficiency by altering the mobility ratio between the displacing slug and displaced oil. Additional oil recovery attained during polymer flooding in the field has been attributed to enhanced sweep efficiency (Moffitt and Mitchell 1983; Greaves et al. 1984; Hochanadel et al. 1990; De Melo et al. 2014; Clemens et al. 2016; Kumar et al. 2016). Polymer solutions do not reduce the IFT significantly. They can increase the N_(c) by 1 or 2 orders maximum with their viscosity, which will not be sufficient to mobilize the significant amount of residual oil. Therefore, conventionally it is believed that polymer flooding cannot reduce S_(or). In recent times, however, viscoelastic polymers were reported to cause an additional residual oil recovery in the Daqing field (Wu et al. 2007). There were some studies which argued against the polymer's viscoelastic influence on the S_(or) reduction under certain conditions (Schneider and Owens 1982; Huh and Pope 2008; Vermolen et al. 2014; Erinick et al. 2018; Seright et al. 2018). Azad and Trivedi (2019a) listed those conditions that are relatively unfavorable for the viscoelastic polymer to recover the residual oil. Nevertheless, several recent results have asserted the polymer viscoelastic influence on S_(or) reduction beyond that of water flood (Wang et al. 2000; Wang et al. 2001; Wang et al. 2007; Xia et al. 2008; Jiang et al. 2008; Wang et al. 2010; Wang et al. 2011; Ehrenfried 2013; Vermolen et al. 2014; Clarke et al. 2015; Qi et al. 2017; Qi et al. 2018). It is to be noted that all the aforementioned literature has asserted the polymer's viscoelastic influence on S_(or) reduction specifically by ensuring the recovered oil is the true residual oil that is well swept. Some studies specifically emphasize the polymer's viscoelastic influence on the S_(or) reduction even at the flux of ˜1 ft/day and the intermediate N_(c) of 10⁻⁵ to 10⁻⁶ (Ehrenfried 2013; Clarke et al. 2016; Cottin et al. 2014; Qi et al. 2017; Koh 2017). The polymer flood shows rapid oil mobilization even before the critical N_(c) (at low flux) which invalidates the capillary theory (Lotfollahi et al. 2016b; Qi et al. 2017). The proper mechanisms causing this S_(or) reduction at low flux are not understood and wettability alteration has been proposed as the mechanism (Seright 2017). Azad and Trivedi (2019b) speculated extensional viscosity might be the reason.

To account for the polymer's sweep efficiency, mobility ratio is widely used (Green and Willhite 1998). Mobility ratio by definition is the ratio between the mobility of displacing slugs and displaced oil. Mobility is the ratio between the permeability and viscosity. The higher the viscosity, the lower the mobility ratio and the higher the sweep efficiency. Viscoelastic polymers possessing higher apparent viscosity contribute more to the higher sweep efficiency than polymers possessing higher elasticity (Azad and Trivedi 2018b). The role of the viscosity of viscoelastic polymers on sweep efficiency has been reported in more literature (Chen et al. 2011; Wang et al. 2013). These clearly indicate the apparent viscosity can be used in the mobility ratio calculation for accounting for the sweep efficiency. However, to account for the microscopic displacement efficiency of viscoelastic polymer flooding, apparent viscosity might not be appropriate, which is discussed below.

The positive influence that the viscoelastic polymers have on residual oil saturation (Sor) reduction during enhanced oil recovery (EOR) has been reported in recent literature. Conventional capillary number (Nc) calculated using the core-scale apparent viscosity is commonly used to correlate the Sor reduction potential of chemical slugs. However, Nc remained the same for the set of polymer solutions of different elasticity, despite the higher Sor reduction shown by the highly elastic polymer solutions. As per the capillary theory, residual oil cannot be mobilized unless the capillary number exceeds the critical capillary number. However, during viscoelastic polymer flooding, significant amount of residual oil was reported to be recovered even before the critical Nc. Therefore, capillary theory fails in the case of viscoelastic polymer flooding with the usage of conventional Nc. Oscillatory Deborah number (De) has been deemed as a better option for correlating the Sor reduction during viscoelastic polymer flooding. High saline viscoelastic polymer solutions possessing lower oscillatory relaxation time were reported to cause higher Sor reduction than low saline viscoelastic polymer possessing higher oscillatory De. Oscillatory De was also reported to be similar for both viscous and viscoelastic polymer solutions. Despite the limitation of conventional Nc and De, all the existing models used for predicting the Sor during viscoelastic polymers relies on either conventional Nc and/or De. Employing those models for predicting Sor in a reservoir simulator could give a false estimate about the oil recovery potential of different viscoelastic polymer solutions. Further, using the oscillatory De or conventional Nc for choosing the optimal polymer for oil recovery applications could lead to erroneous selection. Residual oil recovery is a pore-scale phenomenon, Polymer solutions flowing in the porous media are likely to exhibit 75% non-linear, elongational resistance at pore-scale. Conventional Nc incorporating apparent viscosity may overlook the pore-scale elongational effects and De calculated using the oscillatory rheology may over look the non-linear viscoelastic effects at the pore-scale.

The surfactant solutions having lower IFT (higher N_(c)) were reported to contribute to higher S_(or) reduction (Foster 1973; Karnanda et al. 2013; Sheng 2015). However, higher S_(or) reduction is shown by the highly elastic polymers over viscous or less elastic polymers, despite possessing the similar N_(c) (Qi et al. 2017). It is important to point out here that conventional N_(c) is calculated using the apparent viscosity (Ehrenfried 2013 Qi et al. 2017; Koh 2017; Erinick et al. 2018). Micro-force or normal stress has been identified as the reason for higher S_(or) reduction by highly elastic polymers (Wang et al. 2001; Xia et al. 2004; Xia et al. 2008; Afsharpoor et al. 2012; Wang et al. 2013; Lotfallahi et al. 2016b). Normal stress is related to extensional viscosity (Barnes 2010). The conventional notion that core-scale apparent viscosity encompasses extensional viscosity (Hirasakhi and Pope 1974; Masuda et al. 1992; Delshad et al. 2008; Stavland et al. 2010; Clarke et al. 2015) is refuted by Azad and Trivedi (2019b) who show that actual extensional viscosity is three orders higher than the core-scale apparent viscosity. Azad and Trivedi (2019 b) reported that extensional viscosity should not be treated as the constituent of core-scale apparent viscosity for pore-scale applications such as S_(or) reductions. IFT that measures the tension between the water and oil interface is a microscopic property. Similarly, extensional viscosity which gives the measure of polymer's resistance to stretching around the pore is also a microscopic phenomenon (Haas and Durst 1982). The transient nature of elastic response means that normal stresses or extensional viscosity will be dominated only in the micro region of the pore (Wang et al. 2007). Coincidently, capillary pressure tends to be higher in that micro region characterized by the smaller radius, which in turn causes the higher trapping of residual oil. Microscopic residual oil displacement is a pore-scale phenomenon. So, if the microscopic property such as IFT has been used in the N_(c) for relating it with S_(or) (Green and Willhite 1998), the extensional viscosity should also be incorporated into N_(c). Challenges involved in the extensional measurement of EOR polymers were well documented in previous publications (Azad and Trivedi 2017; Azad et al. 2018a; Azad et al. 2018b; Azad and Trivedi 2018a; Azad and Trivedi 2018b; Azad and Trivedi 2019a).

To quantify the polymer's viscoelastic effects on Sor reduction, few pore-scale models have been proposed. These include models proposed by Chen et al. (2012), Wang et al. (2013), and Lotfallahi et al. (2016b). All these models fail to honor the capillary theory because they either rely on the normal stress and/or Deborah number. All the models that stress the importance of normal stress rely on empirical fitting parameters determined from core flooding. In the case of surfactant flooding, S_(or) reduction can be quantified through IFT, which is a bulk property. It is worthwhile to mention here that the Azad Trivedi viscoelastic model (AT-VEM) is the first viscoelastic model that can predict the apparent viscosity for various ranges of shear rates through bulk rheology alone (Azad and Trivedi 2018a; Azad and Trivedi 2019). Similarly, any methodology that can quantify the S_(or) reduction through bulk properties alone is desirable, which can help in the quick screening of optimal slugs. The model developed by Qi et al. (2018) is independent of core flood experiments. However, Qi et al.'s (2018) method is exclusively based on the conventional Deborah number (D_(e)).

D_(e) is widely used by chemical EOR researchers for quantifying the polymer viscoelastic effects during chemical EOR. If the transit time of the polymer solutions between pore body and pore throat is less compared to their relaxation time, then the fluid will exhibit elastic strain that causes a higher pressure drop, which would be more than expected from shear forces. It is to be noted that the relaxation time used by most EOR researchers are oscillatory based (Magbagbeola 2008; Delshad et al. 2008; Ehrenfried 2013; Vermolen et al. 2014; Koh 2015; Hincapie and Gazner 2015; Qi et al. 2017; Erinick et al. 2018; Qi et al. 2018), which represents the linear viscoelastic effects (Howe et al. 2015). Several misconceptions exist because of the usage of an oscillatory Deborah number, especially when there is variation in the salinity. High saline polymer solutions possessing lower D_(e) were reported to cause higher S_(or) reduction than the low saline polymer solutions possessing higher D_(e) (Ehrenfried 2013; Erinick et al 2018). Magbagbeolo (2008) reported that high saline polymer solutions (with the lower oscillatory Deborah number) resulted in the higher strain hardening index than the low saline polymer solutions (with the higher oscillatory Deborah number). Azad and Trivedi (2018d) also showed, using direct extensional measurements, that high saline polymer solution (with the lower oscillatory D_(e)) provided higher extensional resistance than the low saline polymer solutions (with the higher oscillatory D_(e)). When the polymer solutions flow from the pore body to pore throat, they stretch and generate extensional resistance to flow. Therefore, using the linear relaxation time determined from the oscillatory rheology for mimicking the flow from pore body to pore throat is not ideal for Deborah number calculation (Azad and Trivedi 2019b).

As discussed, N_(c) fails to explain the different residual oil recovery potential of viscoelastic polymers varying in the elasticity. Oscillatory D_(e) appears to be deficient in honoring the non-linear viscoelastic effects that the EOR polymer solutions are expected to exhibit.

Thus, there exists a need for a method of predicting residual oil saturation during viscoelastic polymer flooding during enhanced oil recovery without the above noted limitations, such that polymer flood operators are able to select an optimal polymer before polymer flood when Nc of different viscoelastic polymers remains the same.

SUMMARY OF THE INVENTION

The present invention provides a method of quantifying a viscoelastic effect of a polymer on residual oil saturation (S_(or)). According to embodiments, the polymer is used in polymer flooding is used during enhanced oil recovery (EOR). The method includes calculating an extensional capillary number (N_(ce)) using flux, pore-scale apparent viscosity, and interfacial tension (IFT) to account for the polymer's viscoelastic forces that are responsible for Sor reduction. According to embodiments, an extensional capillary number is calculated for a plurality of polymer materials, which then may be compiled in a database. An increase in the N_(ce) will result in an increase in the S_(or) reduction. Also provided is a reservoir simulator for predicting the S_(or) reduction potential of the viscoelastic polymer, which according to embodiments includes a database of calculated extensional capillary numbers for a plurality of polymers. According to embodiments, the database includes a curve generated from the calculated extensional capillary numbers for a plurality of polymers properties, flux rates, formation nature, oil viscosities, and rheological behaviors.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow-chart containing the steps used in the development of correlation according to embodiments of the present invention;

FIG. 2 is a graph showing the typical extensional viscosity vs the generated strain rate behavior during CaBER experiment;

FIG. 3 is a graph showing N_(ce) vs S_(or) for different data sets;

FIG. 4 is a graph showing extensional CDC generated using the proposed correlations;

FIG. 5 is a plot showing the relation between S_(or) verses N_(c) and N_(ce) of various polymers;

FIG. 6 is a plot showing the relation between S_(or) verses D_(e) and N_(ce) of various polymers, with S_(or) as a function of De for experiment 5 and 6 are marked with a “-” sign;

FIG. 7 is a plot comparing the actual S_(or) values with the values predicted by proposed correlation and Qi et al. (2018)'s correlation for various experiments, with Qi et al's prediction for experiment 15, 16 and 17 are shown with a “-” sign;

FIGS. 8A-8C show graphs of filament diameter vs time plot for EXP 1 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 9A-9C show graphs of Filament diameter vs time plot for EXP 2 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 10A-10C show graphs of Filament diameter vs time plot for EXP 3 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 11A-11C show graphs of Filament diameter vs time plot for EXP 4 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 12A-12C show graphs of Filament diameter vs time plot for EXP 5 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 13A-13C show graphs of Filament diameter vs time plot for EXP 6 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 14A-14C show graphs of Filament diameter vs time plot for EXP 7 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 15A-15C show graphs of Filament diameter vs time plot for EXP 8 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 16A-16C show graphs of Filament diameter vs time plot for EXP 9 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 17A-17C show graphs of Filament diameter vs time plot for EXP 10 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 18A-18C show graphs of Filament diameter vs time plot for EXP 11 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 19A-19C show graphs of Filament diameter vs time plot for EXP 12 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 20A-20C show graphs of Filament diameter vs time plot for EXP 13 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 21A-21C show graphs of Filament diameter vs time plot for EXP 14 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 22A-22C show graphs of Filament diameter vs time plot for EXP 15 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 23A-23C show graphs of Filament diameter vs time plot for EXP 16 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 24A-24C show graphs of Filament diameter vs time plot for EXP 17 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 25A-25C show graphs of Filament diameter vs time plot for EXP 18 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 26A-26C show graphs of Filament diameter vs time plot for EXP 19 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 27A-27C show graphs of Filament diameter vs time plot for EXP 20 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 28A-28C show graphs of Filament diameter vs time plot for EXP 21 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index;

FIGS. 29A-29C show graphs of Filament diameter vs time plot for EXP 22 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index; and

FIGS. 30A-30C show graphs of Filament diameter vs time plot for EXP 23 and the UCM fit to the linear elastic regimes for the determination of relaxation time b) Extensional viscosity as a function of generated strain rate plot showing the sharp rise in the extensional viscosity around the critical Deborah number c) Power law fit to the extensional viscosity vs Hencky strain values around the critical Deborah number for the determination of strain hardening index.

DETAILED DESCRIPTION

The present invention has utility as methods of predicting residual oil saturation during viscoelastic polymer flooding during enhanced oil recovery (EOR). The inventive method allows polymer flood operators are able to select an optimal polymer before polymer flood when Nc of different viscoelastic polymers remains the same. The present invention also provides polymer producers quick tool to analyze designed/manufactured polymers and optimize their polymer design. The present invention provides the N_(c) using the actual measured extensional viscosity (N_(ce)) and then using it for developing a correlation for predicting the S_(or) reduction potential of viscoelastic polymers. Twenty-three different data sets, extracted from different experiments, are used for developing the correlation. The pore-scale in-situ viscosity is presented using the shear and extensional rheological parameters to account for the viscoelastic driving force in the N_(ce). The N_(ce) is compared with the conventional N_(c) and D_(e). The proposed correlation is compared for its predictability with the latest correlation developed at UT-Austin by Qi et al. (2018). It is ensured through comparative discussion that the deficiency persisting with the existing methods are addressed through the N_(ce).

The steps involved in the development of the method to quantify the polymer's viscoelastic effects on S_(or) reduction is shown in the FIG. 1.

It is to be understood that in instances where a range of values are provided that the range is intended to encompass not only the end point values of the range but also intermediate values of the range as explicitly being included within the range and varying by the last significant figure of the range. By way of example, a recited range of from 1 to 4 is intended to include 1-2, 1-3, 2-4, 3-4, and 1-4.

Polymer preparation and CaBER experiments: Capillary breakup extensional rheometer is used to characterize the extensional rheological properties of various polymer solutions. The details about the polymer type, molecular weight, concentration, salinity and temperature are reported in the Table 1. The polymers are obtained from SNF floerger. The polymer solutions are prepared by low speed mixing of 200 rpm. For conducting extensional rheology measurements, small quantity of the prepared polymer solutions is loaded between the two circular plates of 6 mm. The top plate is separated from the bottom plate which result in the formation of filament. The operational conditions during extensional rheological measurements are reported in Table 2. Filament drainage, governed by the balance between the driving capillary action and resisting polymer's viscosity and elasticity, is monitored by a laser micrometer. The filament diameter as a function of time for all the solutions are shown in the FIGS. 8A-30C. Water and glycerin solutions used in Experiment 20 through Experiment 23 possess negligible resistance to break up when compared to the viscoelastic polymer solutions used in Experiment 1 through Experiment 19. Extensional rheological parameters are obtained from the filament diameter data using appropriate theories.

TABLE 1 Shear and extensional rheological properties of various polymer solutions Conc. Salinity Temp μ_(∞) μ_(p) ^(o) λ τ_(ext) μ_(max) EXP Authors Polymer (ppm) (ppm) (° C.) (cP) (cP) (s) n (s) (cP) n₂ 1 Qi et al. HPAM 3630 2100 11000 Room 1 145 0.133 0.632 0.516 620000 3.74 (2017) 2 Qi et al. HPAM 3630 1800 11000 Room 1 110 0.1 0.6 0.352 560000 3.57 (2017) 3 Erinick et al. HPAM 3630 3400 26600 Room 25 232 0.11 0.32 0.456 760000 3.77 (2018) 4 Erinick et al. HPAM 3630 2000 1400 Room 25 232 0.11 0.32 0.25 228000 2.66 (2018) 5 Erinick et al. HPAM 3630 2000 1400 Room 25 232 0.11 0.32 0.25 228000 2.66 (2018) 6 Erinick et al. HPAM 3630 3548 24300 Room 25 232 0.11 0.32 0.44 813000 4.05 (2018) 7 Ehrenfried HPAM 3630 1500 4000 Room 11 139 2 0.81 0.229 410000 3.12 (2013) 8 Ehrenfried HPAM 3630 1500 4000 Room 11 139 2 0.81 0.229 410000 3.12 (2013) 9 Ehrenfried HPAM 3630 1000 1000 Room 11 139 2 0.81 0.117 117000 3.08 (2013) 10 Ehrenfried HPAM 3630 1500 15000 Room 8 56 2 0.86 0.0879 250000 3.39 (2013) 11 Ehrenfried HPAM 3630 1500 15000 Room 8 56 2 0.86 0.0879 250000 3.39 (2013) 12 Ehrenfried HPAM 3630 1500 15000 Room 8 56 2 0.86 0.0879 250000 3.39 (2013) 13 Clarke et al. HPAM 6040 640 4700 Room 19 197 33 0.88 0.19 210000 3.58 (2015) 14 Clarke et al. HPAM 3130 6000 4700 Room 49 197 2.5 0.96 0.0266 40000 3.29 (2015) 15 Koh et al. HPAM 3630 1200 2000 68 4.71 59.78 0.27 0.57 0.307 320000 3.16 (2017) 16 Koh et al. HPAM 3630 1300 2000 68 5.52 156 0.45 0.62 0.37 370000 3.5 (2017) 17 Koh et al. HPAM 3630 2450 2000 68 10.4 1318 1.62 0.62 0.72 620000 3.61 (2017) 18 Koh et al. HPAM 3330 2000 25878 55 6.23 19.68 0.05 0.62 0.24 550000 3.69 (2017) 19 Cottin et al. HPAM 3630 500 5600 65 1.409 23.57 1 0.72 0.082 197000 3.36 (2014) 20 Clarke et al. Water N.A 4700 Room 1 1 N.A N.A 0.00048 173 −2.06 (2015) 21 Clarke et al. Water N.A 4700 Room 1 1 N.A N.A 0.00048 173 −2.06 (2015) 22 Erinick et al. Water N.A 2000 Room 1 1 N.A N.A 0.00048 173 −2.06 (2018) 23 Erinick et al. Glycerin 800000 2000 Room 57 57 N.A N.A 0.001 374 −2.18 (2018) NA: Not available/applicable

TABLE 2 Operational parameters Operational parameters Value Initial distance between top and bottom plate  3 mm Final distance between top and bottom plate 8.2 mm Aspect ratio 2.73

Models for the Extensional Rheological Parameters:

UCM model for determining extensional relaxation time: Extensional relaxation time (τ_(ext)) is attained by fitting the upper convected Maxwell model to the linear part of the filament diameter-time data in the semi-log plot. Extracted and fitted data are represented by blue lines in FIGS. 8A-30C. The slope represents the longest relaxation time (Clasen et al. 2006; Plog et al. 2004 and Azad and Trivedi 2019a) and the average value of the relaxation time is calculated from slope using the Equation 1.

$\begin{matrix} {{D_{mid}(t)} = {{D_{o}\left( \frac{GD_{0}}{4\sigma_{s}} \right)}^{\frac{1}{3}}e^{({{{- t}/3}\tau_{ext}})}}} & {{Equation}\mspace{14mu} 1} \end{matrix}$

where

-   D_(mid)(t)=Midpoint diameter at time t, mm -   D_(o)=Initial Diameter of the sample loaded, mm -   G=Elastic Modulus, Pa -   σ_(s)=Surface tension of polymer samples, mN/m -   τ_(ext)=Extensional relaxation time of sample, s

The extracted relaxation time for all 23 solutions is shown in Table 1. The extensional relaxation time of water is 4*10⁻⁴ s. The extensional relaxation time of glycerin is 1*10⁻³ s. The extensional relaxation time of the viscoelastic polymer solutions is significantly higher than the extensional relaxation time of viscous glycerin (Table 1).

FENE theory for determining maximum extensional viscosity: Extensional viscosity (μ_(ext)) as a function of strain rate, calculated using Equation 2 and Equation 3 for all the data sets are shown in FIG. 8B to 30B.

$\begin{matrix} {\mu_{ext} = {- \frac{\left( {{2X} - 1} \right)\sigma_{s}}{\frac{dD_{mid}}{dt}}}} & {{Equation}\mspace{14mu} 2} \end{matrix}$

where

-   μ_(ext)=Extensional viscosity, Pa·s -   X=Axial correction factor (0.712).

$\begin{matrix} {{\overset{.}{ɛ}(t)} = {{- \frac{2}{D_{mid}(t)}}\left( \frac{d{D_{mid}(t)}}{dt} \right)}} & {{Equation}\mspace{14mu} 3} \end{matrix}$

where

-   {dot over (ε)}=Strain/Elongation rate, s⁻¹

Extensional rheological behavior of the viscoelastic polymer solutions in the shear-free uniaxial extensional field is completely different than the conventional behavior typically observed in the shear field. In the pure-shear field the viscoelastic polymer solutions show a decrease in the viscosity with respect to the imposed strain rate (Delshad et al. 2008; Seright et al. 2011a, b; Azad et al. 2018a,b; Azad and Trivedi 2018a,b; Azad and Trivedi 2019a,b,c). However, the extensional viscosity shows different regimes with respect to the strain rate (Classen 2010; Azad and Trivedi 2019a; Azad and Trivedi 2019b). The extensional viscosity as a function of generated strain rate during uniaxial extensional rheological experiments is shown in the FIG. 2.

It is important to point out here that strain rates are self-selected by the polymer solutions. Initially, the strain rate is high then drops to lower value due the gravitational sagging. This is shown as regime 1 in the FIG. 2. Regime 1 doesn't represent any material property of polymer solutions. Once the filament is formed, the capillary action tends to break the filament. However, the viscosity and elasticity of the polymer solutions tend to resist the capillary action. Initially, the capillary action is resisted by the polymer's viscosity. Since viscosity is a relatively weaker property, the polymer solutions tend to deform which causes the strain rate to increase (Clasen 2010). This is shown as regime 2 in the FIG. 2. Strain rate increases until the elasticity of polymer solutions begins to resist the capillary action. Since the elasticity is a more potent property, strain rate drops and approaches the asymptotic value around the critical Deborah number of 0.66 (Entov and Hinch 1997; Clasen et al. 2006; Clasen 2010). The asymptotic drop in the strain rate to the critical Deborah number results in the sharp increase in the extensional viscosity (Kim et al. 2010; Clasen 2010). This phenomenon is shown as regime 3 in the FIG. 2. The maximum extensional viscosity around the critical Deborah number (μ_(max@Decr)) corresponds to the elastic limit of the polymer solutions (Clasen 2010). The product of critical strain rate and extensional relaxation time corresponds to the critical Deborah number of 0.66. Therefore, the critical strain rate can be found by dividing 0.66 by the polymer's relaxation time. For more details about the extensional viscositystrain rate plot, readers are encouraged to read our earlier publication (Azad and Trivedi 2019a; Azad and Trivedi 2019c). μ_(max@Dec) for all the solutions are reported (Table 1). μ_(max@Dec) is significantly higher for the viscoelastic polymer solutions than glycerin. For example, HPAM 3630 used in Experiment 1 corresponds to the μ_(max@Dec) of 648,000 cP (FIG. 9B and Table 1). The glycerin used in Experiment 23 corresponds to μ_(max@Dec) of only 374 cP (FIG. 30B and Table 1).

Power law theory for strain hardening index: Extensional viscosity (μ; <2) as a function of strain, calculated using Equation 2 and Equation 4 for all the data sets are shown in FIG. 8C to 30C.

$\begin{matrix} {{ɛ(t)} = {2\; {\ln \left( \frac{D_{o}}{D_{mtd}(t)} \right)}}} & {{Equation}\mspace{14mu} 4} \end{matrix}$

where

-   ε=Hencky strain,

The strain hardening index (n₂) gives a measure of the polymer thickening ability in the extensional field. n₂ is determined by using power law fit to the extensional viscosity vs. strain values around the critical D_(e). n₂ is negative for glycerin (Table 1) because it fails to show thickening (FIG. S-23 c). All the viscoelastic polymer solutions show thickening (FIG. 8C to FIG. 26C). Therefore, n₂ for all of them are positive (Table 1). These results clearly imply that elastic solutions possess more resistance than viscous solutions in the extensional field. However, in the oscillatory field, viscous and elastic polymer solutions may possess similar resistance (Garrouch and Gharbi 2006).

S_(or) reduction values from literature data: The correlation relating the S_(or) and N_(ce) during viscoelastic polymer flooding is developed using 23 different data sets. Only the data sets from the polymer flood experiments that are conducted for an extended period for attaining S_(or) are selected. Polymer flood conducted using very low pore volumes of injections are not included. The data sets are chosen only from the tertiary polymer flooding conducted between the flux rate of 0.2 ft/day to 5 ft/day. Most of the experiments are conducted at the flux rate of 1 ft/day. Polymer flood experiments conducted with carbonate formation and micro-model are excluded. S_(or) corresponding to the water flood and glycerin flood is also included. All the experimental and petrophysical details pertaining to the different polymer, glycerin and water flooding can be found in Table 3. S_(or) data corresponding these data sets are also reported in the Table 3. Shear rheological parameters and IFT values, taken from the literature, are also reported in Table 3.

TABLE 3 Petrophysical properties of various polymer flood experiments along with pore-scale and core-scale parameters k μ_(o) v μ_(app-po) μ_(app-co) EXP Authors Formation φ (mD) (cP) (ft/day) (cP) (cP) 1 Qi et al. Bentheimer 0.22 2200 120 0.96 168455 34.8 (2017) Sandstone 2 Qi et al. Bentheimer 0.22 2100 120 0.2 25504 63.4 (2017) Sandstone 3 Erinick et al. Bentheimer 0.24 1480 126 4 493822 25 (2018) Sandstone 4 Erinick et al. Bentheimer 0.24 1480 126 2 54214 39 (2018) Sandstone 5 Erinick et al. Bentheimer 0.25 1480 114 1 29728 56 (2018) Sandstone 6 Erinick et al. Bentheimer 0.25 1480 114 1 243978 52 (2018) Sandstone 7 Ehrenfried Bentheimer 0.23 2398 149 5.28 176117 N.A (2013) Sandstone 8 Ehrenfried Bentheimer 0.23 2125 162 1.06 51033 N.A (2013) Sandstone 9 Ehrenfried Bentheimer 0.23 1597 162 1.07 8893 N.A (2013) Sandstone 10 Ehrenfried Berea 0.18 187 300 1.33 56734 N.A (2013) Sandstone 11 Ehrenfried Berea 0.18 169 300 0.14 7367 N.A (2013) Sandstone 12 Ehrenfried Boise 0.27 475 300 0.91 22259 N.A (2013) Sandstone 13 Clarke et al. Berea 0.23 435 34 1 47527 50 (2015) Sandstone 14 Clarke et al. Berea 0.23 465 34 1 1446 70 (2015) Sandstone 15 Koh et al. Ottawa 0.35 7900 80 1 22308 16 (2017) Sand 16 Koh et al. Ottawa 0.36 6650 120 1 36251 28 (2017) Sand 17 Koh et al. Ottawa 0.37 7311 250 1 109012 108 (2017) Sand 18 Koh et al. Reservoir 0.28 227 72 1 184600 12 (2017) Sand 19 Cottin et al. Sandstone 0.359 2943 7 3 18660 NA  (2014) Reservoir 20 Clarke et al. Berea 0.23 435 34 2 0.971 1 (2015) Sandstone 21 Clarke et al. Berea 0.23 465 34 2 0.972 1 (2015) Sandstone 22 Erinick et al. Bentheimer 0.25 1480 114 4.7 0.96 NA  (2018) Sandstone 23 Erinick et al. Bentheimer 0.25 1480 114 2 56.28 57 (2018) Sandstone σ_(i) dP/L EXP (mN/m) (psi/ft.) N_(ce) N_(c1) N_(c2) D_(e) S_(or) 1 17.3 10 3.29*10⁻² 2.83*10⁻⁵ NA 14.8 0.198 2 17.3 3   1*10⁻³ 8.11*10⁻⁶ NA 0.6 0.31 3 17.3 30.6 0.402  5.8*10⁻⁵ NA 11 0.08 4 17.3 29.1 2.21*10⁻²  5.6*10⁻⁵ NA 100 0.29 5 17.3 23.9 6.06*10⁻³  4.6*10⁻⁵ NA 32 0.32 6 17.3 12.4 4.97*10⁻²  2.4*10⁻⁵ NA 6 0.22 7 25 28.03 1.31*10⁻¹ 6.03*10⁻⁵ 6.47*10⁻⁶ 4.34 0.151 8 25 15.03 7.63*10⁻³ 2.85*10⁻⁵ 1.38*10⁻⁶ 2.18 0.289 9 25 11.51 1.34*10⁻³ 1.64*10⁻⁵ 1.31*10⁻⁶ 72.91 0.297 10 25 83.07 1.06*10⁻² 1.39*10⁻⁵ 7.33*10⁻⁷ 0.38 0.337 11 25 94.73  1.4*10⁻⁴ 1.43*10⁻⁵ 7.88*10⁻⁸ 0.29 0.4 12 25 52.7 2.85*10⁻³ 2.25*10⁻⁵ 6.72*10⁻⁷ 1.46 0.366 13 25 N.A  6.7*10⁻³ NA 7.05*10⁻⁶ 2.2 0.32 14 25 N.A   2*10⁻⁴ NA 9.85*10⁻⁶ 0.021 0.42 15 13.5 N.A  5.8*10⁻³   6*10⁻⁷ NA 2.94 0.26 16 13.5 N.A 9.47*10⁻³  1.7*10⁻⁶ NA 4.2 0.24 17 13.5 N.A  2.8*10⁻²  4.1*10⁻⁶ NA 16 0.23 18 13.5 N.A  4.8*10⁻²  5.5*10⁻⁷ NA 6.5 0.24 19 17.5 N.A  1.1*10⁻²   1*10⁻⁵ NA NA 0.24 20 1 N.A 6.852*10⁻⁶  NA 7.06*10⁻⁶ NA 0.45 21 1 N.A 6.858*10⁻⁶  NA 7.06*10⁻⁶ NA 0.45 22 15.6 15.7 1.02*10⁻⁶ 3.33*10⁻⁵ NA NA 0.45 23 21.3 33.6 1.864*10⁻⁵   5.2*10⁻⁵ NA NA 0.43

Pore scale viscoelastic model: Since 1970s, several core-scale viscoelastic models were developed for predicting the polymer's apparent viscosity (Hirasakhi and Pope 1974; Masuda et al.1992, Delshad et al. 2008). Unified apparent viscosity (UVM), a core-scale model was successfully used to match the viscoelastic polymer's injectivity (Lotfollahi et al. 2015). Another key feature of viscoelastic polymer is their ability to reduce the S_(or). The inability of viscoelastic models to account the reduction in S_(or) at low flux has been reported (Qi et al. 2018; Azad and Trivedi 2019b). Deborah number has been used to account the reduction in S_(or) during viscoelastic polymer flooding at low fluxes (Qi et al. 2017; Qi et al. 2018). Relaxation time attained in the oscillatory shear field is used in the calculation of Deborah number (Qi et al. 2017; Qi et al. 2018). Azad and Trivedi (2019b) highlighted the limitation of using oscillatory relaxation time for quantifying the polymer's viscoelastic effects on S_(or) reduction at saline conditions. At the porescale, polymer solutions are subjected to 75% elongational deformation (Hass and Durst 1984) and it is important to allot similar weightage to extensional viscosity. A model is presented in Equation 5 that can provide an estimate on the polymer's apparent in-situ viscosity at the pore-scale. The input required by this model to predict the pore-scale apparent viscosity are bulk shear rheological parameters, bulk extensional rheological parameters, petrophysical properties such as permeability, porosity and flux rates. Pore-scale apparent viscosity (μ_(app-pore)) for all the experiments, calculated using the Equation 5 is reported in Table 3.

$\begin{matrix} {\mu_{{app} - {pore}} = {\mu_{\infty} + {\left( {\mu_{p}^{o} - \mu_{\infty}} \right)\left\lbrack {1 + \left( {\lambda*\gamma} \right)^{\alpha}} \right\rbrack}^{\frac{({n - 1})}{\alpha}} + {{0.7}5*{\mu_{\max}\left\lbrack {1 - {\exp \left( {- \left( {\beta*\tau_{ext}*\gamma*n_{2}} \right)} \right)}} \right\rbrack}}}} & {{Equation}\mspace{14mu} 5} \end{matrix}$

where

-   μ_(app-pore)=Apparent viscosity at the pore-scale, cP -   μ_(∞)=Infinite shear viscosity, cP -   μ_(p) ^(o) Zero shear viscosity, cP -   λ=Shear characteristic time, s -   γ=Apparent shear rate, s⁻¹ -   α=Correction factor, 2 -   n=Shear thinning index -   μ_(max)=Maximum elongational viscosity around the critical D_(e), cP -   β=Universal constant, 0.01 -   τ_(ext)=Extensional relaxation time, s -   n₂=Strain hardening index

For the same flux rate, μ_(app-pore) are higher for viscoelastic polymers than viscous glycerin. HPAM 3630 used in the experiment 4 corresponds to the μ_(app-pore) of 91,818 cP in 1,480 mD bentheimer sandstone at 2 ft/day (Table 1 and 3). Whereas in experiment 23, glycerin flood conducted at 2 ft/day in1480 mD bentheimer sandstone provides μ_(app-pore) of only 45.37 cP (Table 1 and 3). At 1 ft/day, high Mw HPAM 6040 polymer and low Mw HPAM 3130 polymer used in the experiment 13 and 14 corresponds to the μ_(app-pore) of 68,640 cP and 865 cP respectively at the similar petrophysical conditions (Table 1 and 3). Similarly, in experiment 5 and 6, HPAM 3630 solutions prepared at the salinity of 1400 ppm and 24300 ppm corresponds to the μ_(app-pore) of 28523 cP and 223658 cP respectively at the similar flux rate and petrophysical conditions (Table 1 and 3). These discussions suggest the possibility of higher pore-scale resistance for higher saline viscoelastic polymers solutions compared to low saline solutions, and higher pore-scale resistance for higher Mw polymer solutions.

Extensional Capillary Number

N_(c) can be defined by the ratio of driving viscous force to capillary force (Equation 6). In general, the higher the N_(c), the lower the S_(or). Apparent viscosity is used to represent the viscous force in the conventional N_(c). However, apparent viscosity or conventional N_(c) does not account for the polymer's viscoelastic forces that are responsible for S_(or) reduction at the pore-scale (Azad and Trivedi 2019b). Consequently, polymers of varying elasticity contributed to different S_(or) reduction at the similar N_(c), (Ehrenfred 2013; Qi et al. 2017; Erinick et al. 2018; Azad and Trivedi 2019 b,c). Extensional viscosity of the polymer is responsible for S_(or) reduction at the pore-scale (Azad and Trivedi 2019c) and it is important that the driving viscous force should incorporate extensional resistance. Therefore, a new capillary number N_(ce) is presented in Eq. 7 by replacing the core-scale apparent viscosity with the pore-scale apparent viscosity calculated using Equation 5. N_(ce) for all the experiments calculated using the Equation 7 is also reported in the Table 3.

$\begin{matrix} {N_{c} = \frac{v*\mu_{{app} - {core}}}{\sigma_{i}}} & {{Equation}\mspace{14mu} 6} \\ {N_{ce} = \frac{v*\mu_{{app} - {pore}}}{\sigma_{i}}} & {{Equation}\mspace{14mu} 7} \end{matrix}$

where

-   μ_(app-core)=The core-scale apparent viscosity, which can be     calculated through core flood pressure data, cP -   σ_(i) is the IFT, mN·m⁻¹ -   v is the flux, ft/day

Correlation between N_(ce) and S_(or):

A correlation developed between the oscillatory Deborah number and S_(or) is implemented in the UTCHEM simulator (Qi et al. 2018). The actual S_(or) and S_(or) predicted by Qi et al. (2018)'s correlation for the Data-set 6 are 0.22 and 0.403. At high salinity, oscillatory relaxation time becomes lower (Erinick et al. 2018) which causes the Deborah number to become lower as well. However, strain hardening, an extensional rheological parameter becomes higher for high saline polymer solutions than low saline polymer solutions despite having the lower oscillatory relaxation time (Magbagbeolo 2008). Azad and Trivedi (2019b) provided a detailed critical note on the limitation of using oscillatory relaxation time for quantifying the polymer's viscoelastic effect during EOR. It is important to incorporate extensional rheological parameters over oscillatory rheological parameters while developing a correlation for predicting the S_(or). N_(ce) developed using the pore-scale apparent viscosity (Equation 7) is correlated with the S_(or) values at different conditions. N_(ce) as a function of S_(or) is shown in the FIG. 3. Both consolidated and unconsolidated formations are used (Table 1). The correlations are developed between the N_(ce) and S_(or). As per the classical capillary theory, an increase in the N_(c) will not result in a drastic decrease in the S_(or) up to the critical N_(c) (Stegemeier 1974; Chatzis and Morrow 1984; Peter 2010). After the critical N_(c), the increase in the N_(ce) will result in a significant increase in the S_(or) reduction. Critical N_(c) values were reported between ˜10⁻⁵ and 10⁻⁴ (Chatzis and Morrow 1984; Stegemeier 1974; Abrahams 1975) and are highlighted in FIG. 1. N_(ce) corresponding to the Experiment 18 to Experiment 23 lie on the left side to the critical N_(c). The N_(ce) value corresponding to the Experiment 4 lies closer to critical N_(c). Displacing fluids used in these experiments are either Newtonian water, viscous glycerin, or much less elastic HPAM polymer (Table 1 and 3). There is no significant increase in the S_(or) reduction with the increase in the N_(ce) values up to critical N_(c). The logarithmic fit made from these data is represented by Equation 8. N_(ce) values of the remaining 17 data sets are higher than critical N_(c) values. A clear trend is seen between N_(ce) and S_(or). An increase in N_(ce) value results in a significant decrease in the S_(or) reduction. The relation between the N_(ce) and S_(or) after critical N_(c) is best fitted with the exponential function (Equation 9).

For N_(ce) less than critical N_(c),

S _(or)=−0.007*ln(N _(ce))+0.3523   Equation 8:

For N_(ce) greater than critical N_(c),

S _(or)=0.308*Exp(−3.604*N _(ce))

Using these two fits, a predictive curve for different sets is generated, which looks more like a conventional CDC curve (FIG. 4).

To predict the S_(or) reduction by viscoelastic polymers, only the bulk shear and extensional rheological properties of the polymer are needed. This can aid in the quick screening of optimal polymer for specific reservoir conditions. The curve is generated using different data sets that have a wide variation in polymer properties, flux rates, formation nature, oil viscosities, and rheological behaviors. The proposed correlation can predict the S_(or) for varying range of polymer concentration (500 ppm to 6000 ppm), brine salinity (2000 ppm to 26,000 ppm), temperature (room temperature to 68° C.), flux rates (0.14 ft/day to 5.28 ft/day), permeability (160 mD to 7900 mD), oil viscosity (7 cP to 300 cP), porosity (0.18 to 0.37), different formations (Bentheimer sandstone, Berea sandstone, Boise sandstone, and Ottawa sand pack), different displacing fluids (viscoelastic polymers, viscous glycerin, and Newtonian water).

Extensional Capillary Number vs Conventional Capillary Number

Next, the predictability of N_(ce) is compared with N_(c) (FIG. 5). The main limitation with the conventional N_(c) is that N_(c) values remain the same for the different sets of polymer solutions that show drastic differences in the S_(or) reduction. The ability of polymer to reduce the S_(or) differs based on their elastic nature, molecular weight, and salinity. For example, polymer flooding at the N_(c) of 5.2*10⁻⁵ and flux rate of 1 ft/day in Bentheimer sandstone using viscous glycerin reduced the S_(or) to 0.43, whereas the S_(or) using viscoelastic HPAM 3630 was 0.198. In other cases (Experiments 2 to 9) conducted between the N_(c) of 1*10⁻⁵to 6*10⁻⁵resulted in S_(or) of up to 0.08 with HPAM 3630. S_(or) values of HPAM polymer flood are significantly lower than the S_(or) values of glycerin flood at the similar N_(c) (Table 1 and Table 3). These variation in S_(or) reduction clearly implies that conventional N_(c) cannot explain the relevant mechanisms of S_(or) reduction by the viscoelastic polymers.

Glycerin used in Experiment 23 corresponds to the N_(ce) of 1.5*10⁻⁵. N_(ce) of HPAM 3630 used in Experiment 1 is 6.7*10⁻², which is almost three orders higher than the N_(ce) of glycerin. However, N_(c) of these HPAM 3630 and glycerin are 5*10⁻⁵ and 5.2*10⁻⁵ respectively. Since N_(ce) of glycerin is slightly lower than its N_(c), pore scale apparent viscosity should be slightly lower than the core scale apparent viscosity. Ashfargpour et al. (2012) also reported that core-scale pressure drop is higher than pore-scale pressure drops for the viscous polymers. Therefore, the notion that core scale apparent viscosity encompasses extensional viscosity (Clarke et al. 2016) appears to be true for viscous solutions. Lower N_(ce) values of glycerin also indicate that it does not possess any additional extensional resistance at pore scale which is the reason for its higher S_(or) value of 0.43. However, for viscoelastic HPAM, N_(ce) is higher than N_(c) by three orders which could have given the additional pore-scale extensional resistance needed for mobilizing the residual oil. Also, higher values of N_(ce) for HPAM when compared to its N_(c) indicates that pore-scale apparent viscosity is significantly higher than core-scale apparent viscosity. Similar observation was made by Ashfargpour et al.'s (2012) who reported that pressure drop exhibited by the viscoelastic polymers is higher around the pore scale when compared to pressure drop on the core-scale. Since the pore-scale apparent viscosity is dominated by the extensional resistance, Clarke et al. (2015)'s notion that core scale apparent viscosity encompasses extensional viscosity doesn't seems to be true for viscoelastic polymer solutions.

Furthermore, low saline HPAM solution at N_(c)=5.6*10⁻⁵ and high saline HPAM solution at flooded at N_(c)=2.4*10⁻⁵ resulted in S_(or) of 0.32 and 0.22, respectively (experiment 5 and 6-Table 3). This suggests the ability of higher salinity polymer solution to contribute to significantly lower S_(or) even if their N_(c) values are slightly lower that of the lower salinity HPAM solution. N_(ce) of low saline HPAM 3630 and high saline HPAM 3630 solutions used in the experiments 5 and 6 are 5.8*10⁻³ and 4.56*10⁻² respectively. Higher N_(ce) values shown by high saline polymer solutions suggests they possess relatively higher extensional resistance at the pore scale which lowers the S_(or) significantly.

Similarly, lower Mw HPAM 3130 and higher Mw HPAM 6040 flooded at 1 ft/day in Berea sandstone (experiments 13 and 14) resulted in S_(or) of 0.42 and 0.32, respectively. Their N_(c) values are 9.08*10⁻⁵ and 7.05*10⁻⁵ respectively. Lower residual oil recovery despite higher N_(c) during HPAM 3130 polymer flooding than HPAM 6040 also implies the inadequacy of conventional N_(c). N_(ce) of HPAM 6040 and HPAM 3130 used in the experiments 13 and 14 are 9*10⁻³ and 1*10⁻⁴ respectively. Higher N_(ce) values shown by high Mw HPAM 6040 when compared to low Mw HPAM 3130 indicates the fact it possesses more extensional resistance at the pore scale which explain the lower S_(or) observed during high Mw HPAM 6040 flooding. Clarke et al. (2016)'s concluded in their paper saying that viscoelastic polymers can recover residual oil more than expected from the shear and apparent viscosity. In this paper it is reiterated that extensional viscosity of high Mw polymers causes the significant lowering of S_(or) even when the observed apparent viscosity of low Mw polymer is higher.

Another discrepancy is that oil mobilization is occurring at the N_(c) values of less than 1*10⁻⁵, which is less than the critical N_(c) value of 1*10⁻⁴ (Abrams 1975; Qi et al. 2017). Complete oil mobilization up to S_(or) of less than 0.1 occurs only when the N_(c) value is 10⁻² (Foster 1973; Abrams 1975; Chatzis and Morrow 1984; Jr. et al. 1985). However, HPAM 3630 used in Experiment 3 that resulted in the S_(or) of 0.08 corresponds to the N_(c) and N_(ce) of 5.8*10⁻⁴ and 4.3*10⁻¹, respectively. This indicates that while N_(c) values are lower than the critical N_(c), N_(ce) values are exceeding it. The proper relation between N_(c) and S_(or) is not seen. The best trend one can observe for these data sets has the R₂ value of only 2% to 5%. One cannot use the conventional N_(c) for predicting the viscoelastic polymer's residual oil recovery potential and it therefore cannot be used for screening optimal polymers. The developed correlation using N_(ce) has a R₂ value of 91%. This clearly indicates the N_(ce) is a better method than N_(c) for quantifying the viscoelastic polymer's influence on Sor reduction.

Extensional Capillary Number vs Conventional Deborah Number

Next, N_(ce) is compared with oscillatory D_(e) for predicting behavior of S_(or) reduction during polymer flooding. D_(e) is widely used in the quantification of polymer's viscoelastic effects on the S_(or) reduction (Lotfollahi et al. 2016b; Qi et al. 2017; Erinick et al. 2018; Qi et al. 2018). As can be seen from FIG. 6, there is no proper trend between D_(e) and S_(or). An increase in D_(e) shows the rapid decrease in S_(or) up to a point, after which S_(or) increases rapidly with an increase in D_(e). These discrepancies are mainly caused by the salinity effect. Erinick et al. (2018) performed sequential flooding by injecting low saline polymer solutions and high saline polymer solutions at the same flux rate of 1 ft/day. Erinick et al. (2018) reported that lower salinity HPAM solution yielded S_(or) of 0.32 whereas the higher salinity HPAM solution yielded S_(or) of 0.22 (experiment 5 and experiment 6). However, the D_(e) did not represent this trend. Points corresponding to D_(e) and S_(or) for experiment 5 and 6 are highlighted with a “-” sign in FIG. 6. As can be seen, increasing value of D_(e) doesn't corresponds to the decreasing value of S_(or). N_(c) values for these experiments are similar (Table 3). Therefore, a modified N_(c) is needed. N_(ce) is plotted with S_(or) reduction in FIG. 6. Unlike D_(e), N_(ce) shows a clear trend for all the experimental value of S_(or). As the N_(ce) increases S_(or) decreases. It is important to note that the extensional rheological parameters are higher for the high saline solutions when compared to the low saline solutions (Table 1). Therefore, incorporating extensional rheology through pore-scale viscosity into N_(ce) resulted in higher N_(ce) value (4.5*10⁻²) for higher salinity polymer solution and lower N_(ce) value (5.8*10⁻³) for lower salinity polymer solution. Similarly, in another polymer flooding performed by Erinick et al. (2018) using higher salinity HPAM solution (experiment 3) with D_(e) of 11 at a flux rate of 4 ft/day resulted in the S_(or) of 0.08. However, lower salinity HPAM solution flooding (experiment 4) with D_(e) of 100 at a flux rate of 2 ft/day resulted in S_(or) of 0.29. Higher values of extensional rheological parameters (FIGS. 10B and 11B) instigated the N_(ce) of high saline solutions (4.38*10⁻¹) to be one order higher than low saline solutions (4.2*10⁻²), thereby justifying the S_(or) reduction (Table 3). Ehrenfried (2013) performed two flooding experiments using high saline and low saline viscoelastic polymer solutions (Experiment 7 and Experiment 9). The higher salinity polymer flooding performed at a flux rate of 5.28 ft/day and with a D_(e) of 4.34 resulted in the S_(or) of 0.151. However, the low saline polymer flooding experiments performed at a flux rate of 1.07 ft/day and a much higher D_(e) of 72.9 resulted in the S_(or) of 0.297. Moreover, even after increasing the flux rates to 5.34 ft/day and D_(e) up to 364.54 during the lower salinity polymer flooding no significant oil mobilization occurred and the Sor remained at 0.297. The oscillatory relaxation time of higher salinity polymer solution and lower salinity polymer solution, used in experiment 7 and experiment 9, are 0.11 seconds and 8.9 seconds, respectively (Ehrenfried 2013). Whereas, the extensional relaxation time measured in this work are 0.229 seconds and 0.117 seconds, respectively (Table 1). It is an implication that oscillatory relaxation time of lower salinity polymer solution is highly overestimated. Incorporating extensional parameter and pore-scale viscosity into modified capillary number resulted N_(ce) value of 1.71*10⁻¹ for higher saline polymer solutions is, two orders higher than the N_(ce) of lower salinity polymer solution. These discussions point to the inefficiency of oscillatory rheology, which in turn is the reason for the skewed relation between oscillatory D_(e) and S_(or). On the other hand, N_(ce) determined using the extensional rheological parameter is able to predict the S_(or) reduction caused by the viscoelastic polymer solutions, including those varying in salinity.

Comparison with Qi et al. (2018)'s Correlation

Recently, Qi et al. (2018) proposed a relation between S_(or) and D_(e). The correlation was developed based on the value of D_(e) (Equation 10 and Equation 11).

For D_(e) less than 1,

$\frac{s_{orp}}{s_{orw}} = 1$

For D_(e) greater than 1,

$\frac{s_{orp}}{s_{orw}} = {1 - {{0.1}33*{\log \left( D_{e} \right)}}}$

where

-   S_(orp) is the residual oil saturation to polymer flood -   S_(orw) is the residual oil saturation to water flood.

S_(or) to polymer flood can be predicted using Qi et al.'s (2018) correlation if the S_(orw) and D_(e) are known. The correlation presented in this work (Equation 8 and Equation 9) can also predict the S_(or) to polymer flooding, if N_(ce) is known. Both of these methods do not require any core flood experiments. S_(or) to polymer flood predicted by the proposed correlation and Qi et al.'s (2018) correlation are compared with observed S_(or) values in FIG. 7 and Table 5.

The correlation by Qi et al.'s (2018) over predicts the Sor for the Polymer flooding conducted using high salinity brine (experiment 3, 6 and 7). An opposite behavior is seen during lower salinity polymer flooding in experiment 9 where S_(or) predicted by the Qi et al.'s (2018) correlation is lower than the actual. Since Qi et al.'s (2018) correlation depends on the oscillatory Deborah number it over predicts the elastic effect of low saline solutions and under predicts the elastic effects of high saline solutions. During the extensional rheology performed in this study, the higher salinity polymer solution used in the experiment 3, 6 and 7 shows relatively higher extensional relaxation time, strain hardening index and maximum extensional viscosity than the low saline polymer solutions used in the experiment 4, 5 and 9 (Table 1). Pore-scale apparent viscosity is directly proportional to extensional relaxation time, strain hardening index and maximum extensional viscosity at the critical Deborah number (Equation 5). Therefore, the pore-scale apparent viscosity of high saline solutions with relatively higher extensional resistance is higher than low saline polymer solutions (Table 3). Since N_(ce) incorporates pore-scale apparent viscosity as the driving viscous force, high saline solutions with higher pore-scale resistance corresponded to the higher N_(ce) values than the low saline polymer solutions. Therefore, the proposed correlation overcomes the limitations in Qi et al.'s (2018) correlation to predict the actual S_(or) values at different salinities during polymer flooding. The actual S_(or) values and the values predicted by the proposed correlation are quite actuate at both high and low salinities. One of key findings here is that it is possible to obtain lower S_(or) at high salinity polymer flooding if the polymer extensional properties are higher.

It is also important to note that Qi et al.'s (2018) correlation was developed using experiments conducted on Bentheimer and Berea sandstone. Therefore, prediction of S_(or) during polymer flooding in high permeability sand pack is slightly off using their correlation (experiment 15, 16 and 17). These points are shown as a “-” symbol in FIG. 7. Particularly in experiment 17, there is a notable difference between the S_(or) predicted by the Qi et al.' s (2018) correlation (S_(or)=0.394) and observed S_(or) of 0.23 (Table 3). Besides the high permeability, oil viscosity (250 cP) is relatively very high in experiment 17, resulting in poor sweep during water flooding and high S_(or) because of residual as well as unswept oil. Since Qi et al.'s (2018) correlation uses S_(or) to water flood, the predicted value of S_(or) to polymer flood will also be higher for experiment 17. However, in reality, S_(or) to polymer flood will become lower in the high permeable sand pack due to low capillarity once the viscous fingering is prevented by the polymer flooding. The correlation proposed in this work predicts the S_(or) to polymer flood without having any dependence on the S_(or) to water flood. Therefore, the S_(or) predicted by the proposed correlation for high permeable experiments are comparable with the actual values. For example, S_(or) predicted by the proposed correlation for the experiment 17 is 0.241 which is closer to actual value of 0.23 (Table 3 and FIG. 7).

Accordingly, the present invention provides that the extensional capillary is the first and only version of the capillary number that can be used to quantify the S_(or) reduction caused by the viscoelastic polymer solutions. A comparative prediction is made between the N_(ce), N_(c), and D_(e). The limitation associated with conventional N_(c) and D_(e) is clearly highlighted and a detailed discussion is provided on why the proposed N_(ce) is a better method. Capillary theory considered to be invalidated in the case of viscoelastic polymer flooding is validated using the N_(ce). The correlation developed using the N_(ce) is the first and only method that can predict the S_(or) reduction caused by the viscoelastic polymer solutions through bulk extensional rheology. This will help in choosing the optimal polymer for specific reservoir conditions. The correlations are developed using 23 different data sets. The correlation could predict the S_(or) reduction shown by the viscoelastic polymer solutions after the critical N_(c). The proposed correlations can predict the S_(or) for a varying range of reservoir permeability (169 mD to 7.9 D), porosity (0.18 to 0.37), brine salinity (2000 ppm to 26000 ppm), concentration (500 ppm to 6000 ppm), polymer molecular weight (6 MDa to 35 MDa), flux (0.14 ft/day to 5.8 ft/day), sandstone (benthemier, boise, berea, and sand pack), and oil viscosity (7 cP to 300 cP). For high saline viscoelastic polymer flooding, the proposed correlation has a better S_(or) predictability than Qi et al.'s (2018) correlation. N_(ce) and the proposed correlation can be incorporated into the reservoir simulator for predicting the S_(or) reduction potential of the viscoelastic polymers.

While at least one exemplary embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the described embodiments in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient roadmap for implementing the exemplary embodiment or exemplary embodiments. It should be understood that various changes may be made in the function and arrangement of elements without departing from the scope as set forth in the appended claims and the legal equivalents thereof.

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1. A method of quantifying a viscoelastic effect of a polymer on residual oil saturation (S_(or)), said method comprising: calculating an extensional capillary number (N_(ce)) using $N_{ce} = \frac{v \times \mu_{{app} - {pore}}}{\sigma}$ where v is flux, μ_(app-pore) is pore-scale apparent viscosity, and σ is interfacial tension (IFT) to account for said polymer's viscoelastic forces that are responsible for S_(or) reduction.
 2. The method of claim 1 wherein said polymer is used in polymer flooding is used during enhanced oil recovery (EOR).
 3. The method of claim 1 further comprising calculating an extensional capillary number (N_(ce)) for a plurality of polymer materials.
 4. The method of claim 1 further comprising compiling a database of calculated extensional capillary numbers for a plurality of polymers.
 5. The method of claim 4 wherein the database includes a curve generated from the calculated extensional capillary numbers for a plurality of polymers properties, flux rates, formation nature, oil viscosities, and rheological behaviors.
 6. The method of claim 4 wherein the database is configured to predict the S_(or) for a varying range of polymer concentration, brine salinity, temperature, flux rates, permeability, oil viscosity, porosity, a plurality of formations, a plurality of displacing.
 7. The method of claim 5 wherein the polymer concentration is between 500 ppm to 6000 ppm.
 8. The method of claim 5 wherein the brine salinity is between 2000 ppm to 26,000 ppm.
 9. The method of claim 5 wherein the flux rate is between 0.14 ft/day to 5.28 ft/day.
 10. The method of claim 5 wherein the permeability is between 160 mD to 7900 mD.
 11. The method of claim 5 wherein the oil viscosity is between 7 cP to 300 cP.
 12. The method of claim 5 wherein the porosity is between 0.18 to 0.37.
 13. The method of claim 5 wherein the plurality of formations include any of Bentheimer sandstone, Berea sandstone, Boise sandstone, and Ottawa sand pack.
 14. The method of claim 5 wherein the plurality of displacing fluids include any of viscoelastic polymers, viscous glycerin, and Newtonian water.
 15. The method of claim 1 wherein the flux (v) is between 0.2 ft/day to 5 ft/day.
 16. The method of claim 1 wherein the polymer is subjected to 75% elongational deformation.
 17. The method of claim 1 wherein an increase in the N_(ce) will result in an increase in the S_(or) reduction.
 18. A reservoir simulator for predicting the S_(or) reduction potential of the viscoelastic polymer using the method of claim
 1. 19. The reservoir simulator of claim 18 wherein the reservoir simulator includes a database of calculated extensional capillary numbers for a plurality of polymers.
 20. The reservoir simulator of claim 19 wherein the database includes a curve generated from the calculated extensional capillary numbers for a plurality of polymers properties, flux rates, formation nature, oil viscosities, and rheological behaviors. 